Separation of linear and index covariates in partially linear single-index models

Abstract

Motivated to automatically partition predictors into a linear part and a nonlinear part in partially linear single-index models (PLSIM), we consider the estimation of a partially linear single-index model where the linear part and the nonlinear part involves the same set of covariates. We use two penalties to identify the nonzero components of the linear and index vectors, which automatically separates covariates into the linear and nonlinear part, and thus solves the difficult problem of model structure identification in PLSIM. We propose an estimation procedure and establish its asymptotic properties, which takes into account constraints that guarantee identifiability of the model. Both simulated and real data are used to illustrate the estimation procedure.